Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
1 / 63
The Multiple Flavours of Multilevel Issues for Networks Tom A.B. Snijders
University of Oxford University of Groningen
June 19, 2012
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
2 / 63
Network analysis, certainly multilevel network analysis, is a cooperative venture ...
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
3 / 63
Multiple Flavours
As a statistician, for me originally multilevel analysis was about nested data sets.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
4 / 63
Multiple Flavours
As a statistician, for me originally multilevel analysis was about nested data sets. Then I learned that for sociologists, there is interest in the theoretical distinction between the levels: e.g., pupils in schools exemplify not only multiple populations for which inference sample ⇒ population is important, but also individuals in social contexts, and different sets of actors mutually interacting.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
4 / 63
Multiple Flavours
As a statistician, for me originally multilevel analysis was about nested data sets. Then I learned that for sociologists, there is interest in the theoretical distinction between the levels: e.g., pupils in schools exemplify not only multiple populations for which inference sample ⇒ population is important, but also individuals in social contexts, and different sets of actors mutually interacting. This variety further multiplies when you think of network analysis. . Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
4 / 63
Multiple Flavours
Whose pictures?
Where are we @?
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
5 / 63
Multiple Flavours
Whose pictures?
Where are we @?
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
6 / 63
Multiple Flavours
Whose pictures?
Where are we @?
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
7 / 63
Multiple Flavours
Whose pictures?
Where are we @?
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
8 / 63
Multiple Flavours
Whose pictures?
Where are we @?
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
9 / 63
Multiple Flavours
What is essential for ‘multilevel’ point of view:
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
10 / 63
Multiple Flavours
What is essential for ‘multilevel’ point of view:
Units of different natures;
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
10 / 63
Multiple Flavours
What is essential for ‘multilevel’ point of view:
Units of different natures; these have their own type of influence on variables;
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
10 / 63
Multiple Flavours
What is essential for ‘multilevel’ point of view:
Units of different natures; these have their own type of influence on variables; random/unexplained variability associated with each ‘level’.
.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
10 / 63
Multiple Flavours
levels in one network
1
multiple parallel networks: replication, populations of networks 2
multiple actor sets and multiple types of relation 3
4
Tom A.B. Snijders
large networks
Multilevel Flavours
Manchester Keynote
11 / 63
Multiple Flavours
levels in one network
1
multiple parallel networks: replication, populations of networks 2
multiple actor sets and multiple types of relation 3
4
Tom A.B. Snijders
large networks
Multilevel Flavours
Manchester Keynote
11 / 63
Multiple Flavours
levels in one network
1
multiple parallel networks: replication, populations of networks 2
multiple actor sets and multiple types of relation 3
4
Tom A.B. Snijders
large networks
Multilevel Flavours
Manchester Keynote
11 / 63
Multiple Flavours
levels in one network
1
multiple parallel networks: replication, populations of networks 2
multiple actor sets and multiple types of relation 3
4
Tom A.B. Snijders
e networks
larg
Multilevel Flavours
Manchester Keynote
11 / 63
Levels in One Network
1 One Network, Multiple Levels
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
12 / 63
Levels in One Network
David Krackhardt, in his 2012 Sunbelt Keynote Lecture, talked about the multiple levels of analysis in a network: in a network with N actors, level L is represented by those issues where the number of cases is of order
Tom A.B. Snijders
Multilevel Flavours
NL
.
Manchester Keynote
13 / 63
Levels in One Network
David Krackhardt, in his 2012 Sunbelt Keynote Lecture, talked about the multiple levels of analysis in a network: in a network with N actors, level L is represented by those issues where the number of cases is of order
NL
.
Thus, Level 1 Actors / Nodes
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
13 / 63
Levels in One Network
David Krackhardt, in his 2012 Sunbelt Keynote Lecture, talked about the multiple levels of analysis in a network: in a network with N actors, level L is represented by those issues where the number of cases is of order
NL
.
Thus, Level 1 Actors / Nodes Level 2 Dyads / Ties
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
13 / 63
Levels in One Network
David Krackhardt, in his 2012 Sunbelt Keynote Lecture, talked about the multiple levels of analysis in a network: in a network with N actors, level L is represented by those issues where the number of cases is of order
NL
.
Thus, Level 1 Actors / Nodes Level 2 Dyads / Ties Level 3 Triads
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
13 / 63
Levels in One Network
David Krackhardt, in his 2012 Sunbelt Keynote Lecture, talked about the multiple levels of analysis in a network: in a network with N actors, level L is represented by those issues where the number of cases is of order
NL
.
Thus, Level 1 Actors / Nodes Level 2 Dyads / Ties Level 3 Triads Higher Larger subgroups ∼ hypergraph models
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
13 / 63
Levels in One Network
David Krackhardt, in his 2012 Sunbelt Keynote Lecture, talked about the multiple levels of analysis in a network: in a network with N actors, level L is represented by those issues where the number of cases is of order
NL
.
Thus, Level 1 Actors / Nodes Level 2 Dyads / Ties Level 3 Triads Higher Larger subgroups ∼ hypergraph models Level 0 Network . Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
13 / 63
Levels in One Network
The recognition and distinction of actors and dyads is fundamental for the graph-theoretic basis of network analysis.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
14 / 63
Levels in One Network
The recognition and distinction of actors and dyads is fundamental for the graph-theoretic basis of network analysis. Triads are fundamental for the sociological approach to social networks since Simmel.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
14 / 63
Levels in One Network
The recognition and distinction of actors and dyads is fundamental for the graph-theoretic basis of network analysis. Triads are fundamental for the sociological approach to social networks since Simmel. Hypergraph models are not used so very much. They are a natural representation, e.g., for activities occurring in groups and emails with multiple recipients.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
14 / 63
Levels in One Network
The recognition and distinction of actors and dyads is fundamental for the graph-theoretic basis of network analysis. Triads are fundamental for the sociological approach to social networks since Simmel. Hypergraph models are not used so very much. They are a natural representation, e.g., for activities occurring in groups and emails with multiple recipients. Theoretical interest for the distinction between the actor and dyad levels becomes even more interesting with multivariate networks. . Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
14 / 63
Levels in One Network
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Relations impinge on relations
Multilevel issues in multivariate networks ... on the variety of how relations can affect relations ...
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
17 / 63
Levels in One Network
Relations impinge on relations
Multilevel issues in multivariate networks ... on the variety of how relations can affect relations ...
(cf. also the algebraic approach; e.g., work by Pattison & Breiger.)
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
17 / 63
Levels in One Network
Relations impinge on relations
Multilevel issues in multivariate networks ... on the variety of how relations can affect relations ...
(cf. also the algebraic approach; e.g., work by Pattison & Breiger.) Here the various levels are not nested: ties, dyads, actors, triads, subgroups, ...,
.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
17 / 63
Levels in One Network
Relations impinge on relations
Different relations can impinge on one another in many different ways.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Relations impinge on relations
Different relations can impinge on one another in many different ways. Dyad level direct association (within tie) entrainment
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
18 / 63
Levels in One Network
Relations impinge on relations
Different relations can impinge on one another in many different ways. Dyad level direct association (within tie) entrainment
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
18 / 63
Levels in One Network
Relations impinge on relations
Different relations can impinge on one another in many different ways. Dyad level direct association (within tie) entrainment mixed reciprocity
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
18 / 63
Levels in One Network
Relations impinge on relations
Different relations can impinge on one another in many different ways. Dyad level direct association (within tie) entrainment mixed reciprocity
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
18 / 63
Levels in One Network
Relations impinge on relations
Actor level
mixed popularity
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Relations impinge on relations
Actor level
mixed popularity
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
19 / 63
Levels in One Network
Relations impinge on relations
Actor level
mixed popularity
mixed activity
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
19 / 63
Levels in One Network
Relations impinge on relations
Actor level
mixed popularity
mixed activity
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
19 / 63
Levels in One Network
Relations impinge on relations
Actor level
mixed popularity
mixed activity
mixed twopathity
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
19 / 63
Levels in One Network
Relations impinge on relations
Actor level
mixed popularity
mixed activity
mixed twopathity
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
19 / 63
Levels in One Network
Relations impinge on relations
Triad level
mixed transitive closure
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Relations impinge on relations
Triad level
mixed transitive closure
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
20 / 63
Levels in One Network
Relations impinge on relations
Triad level
mixed transitive closure
agreement ⇒ red tie
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
20 / 63
Levels in One Network
Relations impinge on relations
Triad level
mixed transitive closure
agreement ⇒ red tie
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
20 / 63
Levels in One Network
Relations impinge on relations
Triad level
mixed transitive closure
agreement ⇒ red tie
. . . and other tie orientations . . .
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
20 / 63
Levels in One Network
Relations impinge on relations
This type of cross-network dependencies is discussed for cross-sectional observations in Wasserman & Pattison (1999), with examples in Lazega & Pattison (1999). For longitudinal observations dependencies are multiplied, because we must distinguish between the dependent and the explanatory (antecedent – subsequent) relations.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
21 / 63
Levels in One Network
Multivariate SAOMs
Stochastic Actor-Oriented Models
Dynamics of multivariate networks can be represented by stochastic actor-oriented models as a direct extension of such models for single networks. (Snijders, Lomi & Torlò, Social Networks, in press.) Multivariate dynamics modeled as continuous-time Markov chain, with state = the multivariate network, where tie variables change one by one.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
22 / 63
Levels in One Network
Multivariate SAOMs
This can be extended by dependent behaviour variables and/or two-mode networks. Note that in Markov process modeling, extending the state space means relaxing the Markov assumption: the current state then provides more information.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
23 / 63
Levels in One Network
Example: Vanina’s MBA study
Example Research with Vanina Torlò, Alessandro Lomi, Christian Steglich. International MBA program in Italy; 75 students; 3 waves. 1
Friendship
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
24 / 63
Levels in One Network
Example: Vanina’s MBA study
Example Research with Vanina Torlò, Alessandro Lomi, Christian Steglich. International MBA program in Italy; 75 students; 3 waves. 1
Friendship
2
Advice: To whom do you go for help if you missed a class, etc.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
24 / 63
Levels in One Network
Example: Vanina’s MBA study
Example Research with Vanina Torlò, Alessandro Lomi, Christian Steglich. International MBA program in Italy; 75 students; 3 waves. 1
Friendship
2
Advice: To whom do you go for help if you missed a class, etc.
3
Communication: With whom do you talk regularly.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
24 / 63
Levels in One Network
Example: Vanina’s MBA study
Results will be presented comparing univariate and multivariate models. The difference shows which parts of the rules governing the dynamics of each of the networks are mediated by the other networks.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
25 / 63
Levels in One Network
Example: Vanina’s MBA study
Results: Friendship, univariate – multivariate uni Effect Out-degree Reciprocity Transitive triplets 3-cycles Indegree popularity (p) Outdegree popularity (p) Outdegree activity (p) Sex alter Sex ego Same sex Same nationality Performance alter Performance ego Performance similarity Tom A.B. Snijders
multi
par.
(s.e.)
par.
–1.840 1.604 0.188 –0.095 0.218 –0.383 –0.079 –0.016 –0.158 0.277 0.240 –0.015 –0.076 0.764
(0.233) (0.097) (0.017) (0.030) (0.062) (0.065) (0.041) (0.070) (0.070) (0.065) (0.080) (0.023) (0.024) (0.188)
–4.311 0.756 0.150 –0.065 0.292 –0.237 0.156 –0.061 –0.102 0.144 0.208 –0.017 –0.081 0.367
Multilevel Flavours
(s.e.) (0.395) (0.184) (0.024) (0.037) (0.104) (0.085) (0.044) (0.078) (0.079) (0.078) (0.091) (0.030) (0.025) (0.224)
Manchester Keynote
⇐
← ⇐
←
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Levels in One Network
Example: Vanina’s MBA study
Results: Friendship (across networks) Effect Advice ⇒ Friendship Reciprocal advice ⇒ Friendship Advice ind. ⇒ Friendship popularity Advice outd. ⇒ Friendship activity Communication ⇒ Friendship Reciprocal Comm. ⇒ Friendship Comm. ind. ⇒ Friendship popularity Comm. outd. ⇒ Friendship popularity Comm. outd. ⇒ Friendship activity
Tom A.B. Snijders
Multilevel Flavours
par.
(s.e.)
0.842 0.222 –0.211 –0.143 1.694 0.415 0.112 –0.087 –0.231
(0.256) (0.209) (0.071) (0.077) (0.197) (0.225) (0.096) (0.058) (0.061)
Manchester Keynote
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Levels in One Network
Example: Vanina’s MBA study
Results: Advice, univariate – multivariate uni Effect Out-degree Reciprocity Transitive triplets 3-cycles Indegree popularity (p) Outdegree popularity (p) Outdegree activity (p) Sex alter Sex ego Same sex Same nationality Performance alter Performance ego Performance similarity Tom A.B. Snijders
multi
par.
(s.e.)
par.
–2.267 1.329 0.320 –0.065 0.245 –0.346 –0.088 –0.043 –0.269 0.168 0.450 0.129 –0.107 0.735
(0.321) (0.131) (0.038) (0.061) (0.057) (0.143) (0.062) (0.092) (0.096) (0.086) (0.124) (0.036) (0.034) (0.245)
–5.129 0.314 0.174 –0.103 0.293 0.073 0.150 –0.002 –0.174 0.080 0.371 0.164 –0.071 –0.003
Multilevel Flavours
(s.e.) (0.661) (0.170) (0.045) (0.069) (0.094) (0.225) (0.080) (0.103) (0.102) (0.102) (0.127) (0.047) (0.037) (0.268)
⇐ ⇐
← ←
⇐
Manchester Keynote
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Levels in One Network
Example: Vanina’s MBA study
Results: Advice (across networks)
Effect Friendship ⇒ Advice Reciprocal friendship ⇒ Advice Friendship ind. ⇒ Advice popularity Friendship outd. ⇒ Advice activity Communication ⇒ Advice Reciprocal Comm. ⇒ Advice Comm. ind. ⇒ Advice popularity Comm. outd. ⇒ Advice activity
Tom A.B. Snijders
Multilevel Flavours
par.
(s.e.)
0.670 –0.113 –0.269 –0.199 1.533 0.632 0.100 –0.189
(0.367) (0.210) (0.130) (0.077) (0.339) (0.253) (0.124) (0.077)
Manchester Keynote
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Levels in One Network
Example: Vanina’s MBA study
Communication, univariate – multivariate uni Effect Out-degree Reciprocity Transitive triplets 3-cycles Indegree popularity (p) Outdegree popularity (p) Outdegree activity (p) Sex alter Sex ego Same sex Same nationality Performance alter Performance ego Performance similarity Tom A.B. Snijders
multi
par.
(s.e.)
par.
(s.e.)
–1.116 1.334 0.127 –0.035 0.216 –0.429 –0.057 –0.016 –0.160 0.206 0.009 –0.007 –0.080 0.580
(0.237) (0.068) (0.010) (0.025) (0.043) (0.077) (0.027) (0.058) (0.054) (0.054) (0.061) (0.019) (0.019) (0.129)
–1.977 0.941 0.104 0.006 0.254 –0.481 0.096 –0.012 –0.114 0.139 –0.083 –0.009 –0.048 0.552
(0.433) (0.110) ⇐ (0.011) ← (0.023) (0.063) (0.081) (0.035) ⇐ (0.057) (0.061) (0.055) (0.065) (0.023) (0.023) (0.161)
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Example: Vanina’s MBA study
Results: Communication (across networks) Effect
par.
Friendship ⇒ Comm. Reciprocal friendship ⇒ Comm. Friendship ind. ⇒ Comm. popularity Friendship outd. ⇒ Comm. activity Closure friendship ⇒ Comm. Advice ⇒ Comm. Reciprocal Advice ⇒ Comm. Advice ind. ⇒ Comm. popularity Advice outd. ⇒ Comm. activity Advice ind. ⇒ Comm. activity
Tom A.B. Snijders
Multilevel Flavours
(s.e.)
1.792 –0.011 –0.051 –0.101 –0.127 0.945 0.720 0.027 0.051 –0.127
(0.216) (0.165) (0.055) (0.038) (0.033) (0.256) (0.227) (0.045) (0.053) (0.043)
Manchester Keynote
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Levels in One Network
Example: Vanina’s MBA study
Results: Communication (across networks) Effect Friendship ⇒ Comm. Reciprocal friendship ⇒ Comm. Friendship ind. ⇒ Comm. popularity Friendship outd. ⇒ Comm. activity Closure friendship ⇒ Comm. Advice ⇒ Comm. Reciprocal Advice ⇒ Comm. Advice ind. ⇒ Comm. popularity Advice outd. ⇒ Comm. activity Advice ind. ⇒ Comm. activity
Tom A.B. Snijders
Multilevel Flavours
par.
(s.e.)
1.792 –0.011 –0.051 –0.101 –0.127 0.945 0.720 0.027 0.051 –0.127
(0.216) (0.165) (0.055) (0.038) (0.033) ⇐ (0.256) (0.227) (0.045) (0.053) (0.043)
Manchester Keynote
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Levels in One Network
Example: Vanina’s MBA study
Dyad level F
Dyad level
C
A All relations positively associated within dyads, direct effects stronger than reciprocal effects.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Example: Vanina’s MBA study
Actor level, incoming F Actor level: incoming ties (popularity) (red = negative association)
C
A At actor-level for incoming ties: friendship and advice negatively associated, positive association communication ⇒ friendship.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Example: Vanina’s MBA study
Actor level, outgoing F Actor level: outgoing ties (activity) (red = negative association)
C
A At actor-level for outgoing ties: all relations weakly negatively related; A → C is mixed effect: incoming advice ties lead to less comm. activity. Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Example: Vanina’s MBA study
Triad level
negative mixed closure Friendsh. ; Comm.
F
F
C Note that this is an effect additional to the positive triadic closure of friendship and the direct dyadic effect of friendship on communication. This will be taken up below.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Example: Vanina’s MBA study
Conclusions: levels
Dyad level and actor level tell different stories.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Example: Vanina’s MBA study
Conclusions: levels
Dyad level and actor level tell different stories. Individuals specialize: learning ⇔ sociability
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
36 / 63
Levels in One Network
Example: Vanina’s MBA study
Conclusions: levels
Dyad level and actor level tell different stories. Individuals specialize: advice ⇔ friendship
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
36 / 63
Levels in One Network
Example: Vanina’s MBA study
Conclusions: levels
Dyad level and actor level tell different stories. Individuals specialize: advice ⇔ friendship Tied dyads tend to have multiplex ties: advice and friendship.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
36 / 63
Levels in One Network
Example: Vanina’s MBA study
Conclusions: dyad level Cross-dependencies between networks change the representation of the internal dynamics: Observed reciprocation and homophily for networks are partly accounted for by multiplex entrainment and reciprocation & homophily in the other networks; this is the case especially for advice;
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
37 / 63
Levels in One Network
Example: Vanina’s MBA study
Conclusions: dyad level Cross-dependencies between networks change the representation of the internal dynamics: Observed reciprocation and homophily for networks are partly accounted for by multiplex entrainment and reciprocation & homophily in the other networks; this is the case especially for advice; homophily in friendship mediated by communication; performance homophily remains only for communication; homophily in advice remains only for nationality.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
37 / 63
Levels in One Network
Example: Vanina’s MBA study
Conclusions: dyad level Cross-dependencies between networks change the representation of the internal dynamics: Observed reciprocation and homophily for networks are partly accounted for by multiplex entrainment and reciprocation & homophily in the other networks; this is the case especially for advice; homophily in friendship mediated by communication; performance homophily remains only for communication; homophily in advice remains only for nationality. Communicate with those having similar performance, and they will advise you. Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Example: Vanina’s MBA study
Conclusions: triad level Most cross-network triadic effects disappear when controlling for actor-level dependencies. This illustrates/generalizes Feld’s (ASR 1982) remark that unmodeled degree heterogeneity may lead to spurious conclusions of transitivity. The only remaining cross-network triadic effect is the negative mixed closure Friendship ; Communication,
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Example: Vanina’s MBA study
Conclusions: triad level Most cross-network triadic effects disappear when controlling for actor-level dependencies. This illustrates/generalizes Feld’s (ASR 1982) remark that unmodeled degree heterogeneity may lead to spurious conclusions of transitivity. The only remaining cross-network triadic effect is the negative mixed closure Friendship ; Communication, counteracting the positive dyadic dependency F ⇔ C; they communicate relatively less with friends of friends who are not also direct friends.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Levels in One Network
Network Theory
Network Theory Pattern ??? There are two main networks (friendship and advice), representing interactions that are connected to two distinct goals: social well-being and academic success. The two goals are not contradictory but pursuing them demands time – a limited resource.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
39 / 63
Levels in One Network
Network Theory
Network Theory Pattern ??? There are two main networks (friendship and advice), representing interactions that are connected to two distinct goals: social well-being and academic success. The two goals are not contradictory but pursuing them demands time – a limited resource. At the dyadic level multiplexity is dominant, leading to positive association between the networks, related to multi-functionality of personal contacts; at the actor level, specialization is dominant (depending on preferences and comparative advantage), leading to negative association between the networks. . Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
39 / 63
Levels in One Network
Network Theory
Here only networks are considered; it would be interesting to combine this with individual goal achievement results.
.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Multiple Parallel Networks
2 Multiple Parallel Networks
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Multiple Parallel Networks
Sample from a population of networks
Multilevel network analysis in the sense of analyzing multiple similar networks, mutually independent, permits research to transcend the level of network as case studies, and to generalize to a population of networks. This was proposed by Snijders & Baerveldt (J. Math. Soc. 2003). Also see Entwisle, Faust, Rindfuss, & Kaneda (AJS, 2007).
.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Multiple Parallel Networks
Sample from a population of networks
Sample from Population of Networks Suppose we have a sample indexed by j = 1, . . . , N from a population of networks, where the networks are similar in some sense; stochastic replicates of each other;
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Multiple Parallel Networks
Sample from a population of networks
Sample from Population of Networks Suppose we have a sample indexed by j = 1, . . . , N from a population of networks, where the networks are similar in some sense; stochastic replicates of each other; they all are regarded as realizations of processes obeying the same model, but having different parameters θ1 , . . . , θj , . . . , θN .
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Multiple Parallel Networks
Fixed effects
Meta-Analysis ∼ Fixed Effects Model: θ1 , . . . , θj , . . . , θN are arbitrary values, no assumption about a population is made.
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Fixed effects
Meta-Analysis ∼ Fixed Effects Model: θ1 , . . . , θj , . . . , θN are arbitrary values, no assumption about a population is made. two-stage procedure: estimate each θj separately, combine the results by Fisher’s procedure for combining independent tests: ‘is there any evidence for a hypothesized effect?’
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Multiple Parallel Networks
Fixed effects
Meta-Analysis ∼ Fixed Effects Model (contd.): For coordinate k of the parameter, test null hypothesis H0 : θkj = 0 for all j against alternative hypothesis H1 : θkj = 0 for at least one j . (Two-sided variants also are possible.) Procedure: see, e.g., Snijders & Bosker Section 3.7.
Mercken, Snijders, Steglich, & de Vries (2009) applied this in a study of smoking initiation: 7704 adolescents in 70 schools in 6 countries. . Tom A.B. Snijders
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Multiple Parallel Networks
Random effects: two-stage procedure
Meta-Analysis ∼ Random Effects Model: θ1 , . . . , θj , . . . , θN are drawn randomly from a population P[net] of networks, no further distributional assumptions are made.
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Multiple Parallel Networks
Random effects: two-stage procedure
Meta-Analysis ∼ Random Effects Model: θ1 , . . . , θj , . . . , θN are drawn randomly from a population P[net] of networks, no further distributional assumptions are made. Two-stage procedure: estimate each θj separately, combine the results in a meta-analysis (Cochran 1954), (‘V-known problem in multilevel analysis) which allows testing hypotheses about P[net] such as, for a coordinate k , Htotal : 0 mean H0 : spread
H0
all θkj = 0; E{θkj } = 0; : var{θkj } = 0. .
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Multiple Parallel Networks
Random effects: two-stage procedure
The input for the meta-analysis consists of estimates θˆj and their standard errors s.e.j . The meta analysis is constructed based on the model θˆj = μ + Uj + Ej , where μ is the population mean, Uj is the true effect of group j, and Ej is the statistical error of estimation.
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Multiple Parallel Networks
Random effects: two-stage procedure
The input for the meta-analysis consists of estimates θˆj and their standard errors s.e.j . The meta analysis is constructed based on the model θˆj = μ + Uj + Ej , where μ is the population mean, Uj is the true effect of group j, and Ej is the statistical error of estimation. Uj and Ej are independent residuals with mean 0, the Uj are i.i.d. with unknown variance, and var(Ej ) = s.e.2j (‘V–known’). . Tom A.B. Snijders
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Multiple Parallel Networks
Random effects: two-stage procedure
Meta-Analysis ∼ Random Effects Model (contd.) This has been applied in quite many studies: e.g., Lubbers (2003): homophily in 57 classrooms with 1466 students (also with integrated random coefficient p∗ approach); Baerveldt, van Duijn, Vermeij, van Hemert (2004): ethnic homophily in 20 schools, 1317 students; Valente, Fujimoto, Chou, and Spruit-Metz (2009): friendship & obesity, 17 classrooms with 617 students; Mercken, Snijders, Steglich, & de Vries (2009): fr. & smoking, 7704 adolescents, 70 schools, 6 countries; also other studies by Liesbeth Mercken. . Tom A.B. Snijders
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Multiple Parallel Networks
Random effects: integrated procedure
Meta-Analysis ∼ Integrated Random Effects Model: θ1 , . . . , θj , . . . , θN are drawn randomly from a population P[net] of networks, and are assumed to have a common multivariate normal N (μ, Σ) distribution, perhaps conditionally on network-level covariates.
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Multiple Parallel Networks
Random effects: integrated procedure
Meta-Analysis ∼ Integrated Random Effects Model: θ1 , . . . , θj , . . . , θN are drawn randomly from a population P[net] of networks, and are assumed to have a common multivariate normal N (μ, Σ) distribution, perhaps conditionally on network-level covariates. Integrated procedure: Estimate μ and Σ and consider the ‘posterior’ distribution of θj given the data.
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Multiple Parallel Networks
Random effects: integrated procedure
Meta-Analysis ∼ Integrated Random Effects Model: θ1 , . . . , θj , . . . , θN are drawn randomly from a population P[net] of networks, and are assumed to have a common multivariate normal N (μ, Σ) distribution, perhaps conditionally on network-level covariates. Integrated procedure: Estimate μ and Σ and consider the ‘posterior’ distribution of θj given the data. Advantage: The analysis of the separate networks draws strength from the total sample of networks by regression to the mean. Useful especially for many small networks. Tom A.B. Snijders
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Multiple Parallel Networks
Random effects: integrated procedure
Meta-Analysis ∼ Integrated Random Effects Model (contd.) New developments in collaboration between Johan Koskinen and T.S., for the stochastic actor-oriented model for network dynamics. Recall that this is a model for network dynamics, where the dynamics is an unobserved sequence of ‘micro steps’ and the parameters are estimated from network panel data.
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Multiple Parallel Networks
Random effects: integrated procedure
Meta-Analysis ∼ Integrated Random Effects Model (contd.) New developments in collaboration between Johan Koskinen and T.S., for the stochastic actor-oriented model for network dynamics. Recall that this is a model for network dynamics, where the dynamics is an unobserved sequence of ‘micro steps’ and the parameters are estimated from network panel data. This is elaborated following a likelihood-based approach; see Koskinen & Snijders (JSPI 2007), Snijders, Koskinen & Schweinberger (AAS 2010), Schweinberger (PhD thesis 2007, Chapters 4 and 5). . Tom A.B. Snijders
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Random effects: integrated procedure
Here we discuss a Bayesian approach, where the parameters μ, Σ have a prior distribution. We assume the conjugate prior, Σ−1 s wishartp (Λ0−1 , ν0 ), and conditionally on Σ μ | Σ s Np (μ0 , Σ/ κ0 ) . Thus, the parameters of the prior are Λ0 , ν0 , κ0 .
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Multiple Parallel Networks
Random effects: integrated procedure
The joint p.d.f., for data y1 , . . . , yj , . . . , yN , is
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Random effects: integrated procedure
The joint p.d.f., for data y1 , . . . , yj , . . . , yN , is fInvWish Σ | Λ0−1 , ν0 ϕp μ | (μ0 , Σ/ κ0 )
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prior
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Multiple Parallel Networks
Random effects: integrated procedure
The joint p.d.f., for data y1 , . . . , yj , . . . , yN , is fInvWish Σ | Λ0−1 , ν0 ϕp μ | (μ0 , Σ/ κ0 ) QN × j=1 ϕp (θj | μ, Σ)
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prior hierarchical model
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Multiple Parallel Networks
Random effects: integrated procedure
The joint p.d.f., for data y1 , . . . , yj , . . . , yN , is fInvWish Σ | Λ0−1 , ν0 ϕp μ | (μ0 , Σ/ κ0 ) QN × j=1 ϕp (θj | μ, Σ)
×
QN j=1
pSAOM (yj | θj )
prior hierarchical model network model
Since pSAOM (yj | θj ) cannot be calculated directly, we employ data augmentation (Tanner & Wong, 1987): augment the network panel data by the sequence vj of all microsteps connecting the consecutive observations. . Tom A.B. Snijders
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Multiple Parallel Networks
Random effects: integrated procedure
The joint p.d.f., for data y1 , . . . , yj , . . . , yN , using data augmentation, is the sum over vj of
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Multiple Parallel Networks
Random effects: integrated procedure
The joint p.d.f., for data y1 , . . . , yj , . . . , yN , using data augmentation, is the sum over vj of fInvWish Σ | Λ0−1 , ν0 ϕp μ | (μ0 , Σ/ κ0 ) prior QN hierarchical model × j=1 ϕp (θj | μ, Σ) ×
QN j=1
pSAOM (vj | θj , yj ) .
Tom A.B. Snijders
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network model
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Multiple Parallel Networks
Random effects: integrated procedure
The joint p.d.f., for data y1 , . . . , yj , . . . , yN , using data augmentation, is the sum over vj of fInvWish Σ | Λ0−1 , ν0 ϕp μ | (μ0 , Σ/ κ0 ) prior QN hierarchical model × j=1 ϕp (θj | μ, Σ) ×
QN j=1
pSAOM (vj | θj , yj ) .
network model
The posterior distribution can be sampled by Markov chain Monte Carlo (MCMC). The unknown random variables are μ, Σ; θ1 , . . . , θN ; v1 , . . . , vN and these are sampled in turn, as follows. . Tom A.B. Snijders
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1
Random effects: integrated procedure
For all j make some Metropolis Hastings steps sampling vj | yj , θj , as in Snijders, Koskinen & Schweinberger (2010).
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Multiple Parallel Networks
Random effects: integrated procedure
1
For all j make some Metropolis Hastings steps sampling vj | yj , θj , as in Snijders, Koskinen & Schweinberger (2010).
2
For all j make one or more Metropolis Hastings steps sampling θj | vj , μ, Σ, using a random walk proposal distribution (Schweinberger 2007, Ch. 5.4; Koskinen & Snijders 2007, Sect. 4.4).
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Multiple Parallel Networks
Random effects: integrated procedure
1
For all j make some Metropolis Hastings steps sampling vj | yj , θj , as in Snijders, Koskinen & Schweinberger (2010).
2
For all j make one or more Metropolis Hastings steps sampling θj | vj , μ, Σ, using a random walk proposal distribution (Schweinberger 2007, Ch. 5.4; Koskinen & Snijders 2007, Sect. 4.4).
3
Sample (μ, Σ) | θ1 , . . . , θN , Λ0 , ν0 , κ0 from the full conditional distribution.
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Multiple Parallel Networks
Random effects: integrated procedure
1
For all j make some Metropolis Hastings steps sampling vj | yj , θj , as in Snijders, Koskinen & Schweinberger (2010).
2
For all j make one or more Metropolis Hastings steps sampling θj | vj , μ, Σ, using a random walk proposal distribution (Schweinberger 2007, Ch. 5.4; Koskinen & Snijders 2007, Sect. 4.4).
3
Sample (μ, Σ) | θ1 , . . . , θN , Λ0 , ν0 , κ0 from the full conditional distribution.
This requires tuning to obtain good mixing – as usual. . Tom A.B. Snijders
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Multiple Actor Types
3 Multiple Actor Sets and Multiple Relations
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Multiple Actor Types
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Multiple Actor Types
Multilevel network analysis with multiple actor types was pioneered by Breiger (Soc. Forces, 1974), Hedstrøm, Sandell & Stern (AJS, 2000) and Lazega, Jourda, Mounier & Stofer (Social Networks, 2008).
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Multiple Actor Types
Multilevel network analysis with multiple actor types was pioneered by Breiger (Soc. Forces, 1974), Hedstrøm, Sandell & Stern (AJS, 2000) and Lazega, Jourda, Mounier & Stofer (Social Networks, 2008). The 2012 Sunbelt Conference saw an avalanche of new work, with contributions by Alessandro Lomi, Peng Wang, and Johan Koskinen. At this conference we see many further developments.
Tom A.B. Snijders
Multilevel Flavours
Manchester Keynote
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Multiple Actor Types
Multilevel network analysis with multiple actor types was pioneered by Breiger (Soc. Forces, 1974), Hedstrøm, Sandell & Stern (AJS, 2000) and Lazega, Jourda, Mounier & Stofer (Social Networks, 2008). The 2012 Sunbelt Conference saw an avalanche of new work, with contributions by Alessandro Lomi, Peng Wang, and Johan Koskinen. At this conference we see many further developments. what can I add???
Tom A.B. Snijders
Multilevel Flavours
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Multiple Actor Types
Multilevel network analysis with multiple actor types was pioneered by Breiger (Soc. Forces, 1974), Hedstrøm, Sandell & Stern (AJS, 2000) and Lazega, Jourda, Mounier & Stofer (Social Networks, 2008). The 2012 Sunbelt Conference saw an avalanche of new work, with contributions by Alessandro Lomi, Peng Wang, and Johan Koskinen. At this conference we see many further developments. what can I add???
Longitudinal data of linked networks with multiple actor types also can be analyzed using RSiena, thanks to the flexible design by Krists Boitmanis and Ruth Ripley. Tom A.B. Snijders
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Large Networks
4 Large Networks
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Large Networks
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Large Networks
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Large Networks
A collection of many groups ∼ many ‘parallel’ networks is really a case of a large network where between-group ties are ignored. The applicability of our usual network models to large groups (hundreds of actors and more) seems limited by the fact that we still do not know a lot about how the large-scale structure of networks differs from the small-scale and medium-scale structures; and large networks must be full of heterogeneity where, e.g., ERGM parameters will not be constant in all ‘regions’ of the network – but how to define such ‘regions’? . Tom A.B. Snijders
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Large Networks
Models for large networks should incorporate latent heterogeneity – at which level?
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Large Networks
Models for large networks should incorporate latent heterogeneity – at which level? actors / subgroups ?
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Large Networks
Models for large networks should incorporate latent heterogeneity – at which level? actors / subgroups ? Work by Johan Koskinen – Michael Schweinberger – ....
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the road
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on and on
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